Estimating the correlation dimension of a fractal on a sphere

The correlation dimension is a fundamental parameter in order to characterize fractal geometries. The estimation of the correlation dimension on a spherical manifold is of major interest in many applications, especially in climate science and geophysics. In this paper we investigate the use of the Takens estimator of the correlation dimension on a sphere. By relying on two analytically tractable cases, we show how using geodetic and Euclidean metrics ensues different kinds of bias. As a major application, our analysis provides a cue for investigations of fractal geometries in seismology.